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Argumentation Frameworks with Higher-Order Attacks: Semantics and Complexity
Sylvie Doutre, Mickaël Lafages, Marie-Christine Lagasquie-Schiex
To cite this version:
Sylvie Doutre, Mickaël Lafages, Marie-Christine Lagasquie-Schiex. Argumentation Frameworks with Higher-Order Attacks: Semantics and Complexity. 17th International Conference on Principles of Knowledge Representation and Reasoning, Sep 2020, Rhodes (virtual conference), Greece. 2020. �hal-02942437�
Argumentation Frameworks with Higher-Order Attacks:
Semantics and Complexity
Sylvie Doutre, Mickaël Lafages, Marie-Christine Lagasquie-Schiex
IRIT, University of Toulouse, Toulouse, France {sylvie.doutre, mickael.lafages, lagasq}@irit.fr
Introduction
Argumentation frameworks (AF) are formalisms to express argumentation
problems. In Dung’s one, they are expressed as directed graph in which nodes represent argument and arrow, attack relations between arguments. Higher-order frameworks, unlike Dung’s one, allow to have attacks over attacks. RAF are such a framework (see Figures 1 and 2). Arguments are here represented by circles and attack relations by squares.
For future algorithm investigations, we adapted the notion of Dung’s AF labellings for RAF. We showed the relation between structures (counterpart of extensions for RAF) and different types of structure labellings.
We studied the complexities of RAF decisions problems and shown that despite the higher expressiveness offered by them, the decision classes stay the same as Dung’s AF.
Figure 1 Figure 2
Examples of RAF
Complexities of RAF
We introduced a new flattening of RAF to Dung’s AF (procedure called Raf2Af) in order to prove that it is also the case for RAF complexities.
Table 3 summarises the complexities of the credulous and skeptical acceptance problems, the verification, the existence, the non-empty existence and the uniqueness problems.
Figure 3 shows an example of flattening. For each attack two arguments are created : one, named as the attack, representing the validity of the attack, the other one the validity of both the attack and its source. For each argument, an other one is created representing the invalidity of the argument.
Same complexities as Dung’s AF
Figure 3 : Raf2AF(Γ) of Figure 2 RAF
Labellings for RAF
Instead of extensions (set of arguments), RAF solutions are expressed as
structure: a couple of sets, one of arguments and one of attacks. As for
Dung’s AF, we introduced structure labellings for RAF, a couple of labellings, one for the arguments and the other one for the attacks. They are three value-based: in (accepted), out (rejected), und (undecidable).
Reinstament RAF labellings are particular labellings that coïncide under
some constraints to differents RAF semantics (see Tables 1 and 2).
Table 1: RAF labellings for Figure 1
Table 3: Complexities for RAF decision problems
Perspectives
• Algorithms for RAF argumentation problems
• Complexities of function problems
